Lost records stock purchase date?

I have a problem. I have lost a brokerage account and their statement
showing my cost basis for a stock.
I know that sometime between 10/20/04 and 12/29/04, I purchased 500 shares
of stock, but I don't know what brokerage was used. I've able to pin down
the dates because I closed out one brokerage account on 10/20/04. On
12/29/04 I transferred the new shares into a 3rd account, but don't
know where the shares were for approximately 2 months. The incoming
transfer came from Pershing, but that seems to be a clearing house for
several brokerages.

Since I know the time period when the stock was purchased,
(10/20/04-12/29/04) would the IRS accept the average closing price
for that date range as a cost basis?

This is very vexing to me because I'm a pack rat and have records going back
to 1970, I just can't find this one.

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